https://file.aiquickdraw.com/mathbot/file/1726086354564fxtwvdoa.jpg

The function provided in your image is a logistic growth model:

\[
Z(t) = \frac{315}{1 + 14e^{-0.23t}}
\]

Where:
- \( Z(t) \) represents the zombie population (in thousands),
- \( t \) is time in days,
- The constant values are \( 315 \) (limiting population) and \( 14 \) (growth factor).

### Graphing the function
To visualize this, you can plot the function over the domain (0, 70) using a graphing tool like Desmos or any other graphing calculator.

### Steps to Graph the Function:
1. **Input the function**: 
   \[
   Z(t) = \frac{315}{1 + 14e^{-0.23t}}
   \]
2. **Set the domain**: 
   Since zombies are produced for 70 days, your domain for \( t \) should be from \( t = 0 \) to \( t = 70 \).
3. **Set the range**: 
   The population approaches 315 (thousand) as \( t \) increases. So the range should be approximately \( Z(t) \in [0, 315] \).

### Shape of the Graph:
- The curve will start near \( Z(0) \approx 0 \) and grow quickly initially.
- As \( t \) increases, the rate of growth slows down, and \( Z(t) \) asymptotically approaches 315, forming a classic "S-curve" shape characteristic of logistic growth models.

I recommend plotting the function using Desmos for a clear visualization.

### Limiting Value:
The limiting value of \( Z(t) \), as \( t \to \infty \), is approximately 315 (thousand). This is the maximum zombie population predicted by the model.

Would you like further details on how to estimate other values, or questions about the graph?

---

Here are 5 related questions:
1. How do you interpret the meaning of the limiting value of \( 315 \) in real-world terms?
2. What does the factor \( 14e^{-0.23t} \) represent in the logistic growth equation?
3. How would changing the constants (e.g., 315 or 0.23) affect the shape of the curve?
4. What is the significance of the exponential decay in logistic growth?
5. How could you estimate the point where the growth rate of \( Z(t) \) is at its maximum?

**Tip**: When working with logistic functions, remember the midpoint (where growth is fastest) occurs when the population is half of the limiting value.

This problem explores a logistic growth model that simulates the population growth of zombies after a nuclear fallout. Using the formula Z(t) = 315 / (1 + 14e^(-0.23t)), students will graph the function over 70 days to observe how the population approaches 315 (thousand). Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation